# Math intro courses guide

### Welcome to math at Yale!

This site is for incoming first-year, transfer, and Eli-Whitney undergraduates, as well as for all other students who are looking for advice about their first math courses at Yale. We look forward to working with you!

Whether you plan to take one course, several, or are considering a mathematics major, we are here to support you. This page will help you get started.

You can also take a look at our orientation video, which provides basic information about our courses - and it has a puzzle game at the end!

If you are thinking about the math major or would like to learn more about the department, we encourage you to sign up for our Math DUS mailing list, take a look at our Math community page or explore other tabs on the left that interest you.

### Announcements

- May 16: The 2024 Yale placement exams will open at 9am ET on July 1, and close at 12pm (noon) on July 31.
- May 16: We will update this site with the rest of 2024 information by the end of May.

#### Summer / Fall 2023 event schedule

**1. This year’s calculus placement exam will be open from 9am on July 3 to 12pm (noon) on July 31**. In general, opening dates of all placement exams are listed on the following Yale College site. Details about the math placement test can be found below.

At the beginning of August, each calculus course will post a short video with basic information about the class, tips for success in the course, and other relevant information.

2. Links to short videos for each of the main coures will be posted here during the first week of August.

**3. Individual calculus placement advising session **schedule is below. If you need placement advice, wait to submit your preference selection entry until after the advising session (you will have several more days to submit, and the timing of your entry does not affect the lottery outcome in any way).

Please note that to receive advice, you must first complete the placement exam - advisers cannot give you placement without it.

**Math 110 individual placement advising session** will take place on Tuesday, August 8, 1:20 - 2:30pm ET via zoom. This sessions is specifically for students who placed into Math 110 and seek placement advice, or who placed elsewhere and wish to take Math 110 instead. Click here to join the session.

**All other students** who seek placement advice should attend one of the following individual placement advising sessions:

- Wednesday, August 9, 7:20 - 9pm ET. Click here to join the session.
- Thursday, August 10, 2:20 - 4pm ET. Click here to join the session.

**A makeup placement advising session**, for all students who need assistance with their calculus placement, will take place on *Sunday, August 27, 2:20 - 4pm, in person. It will be at 17 Hillhouse Avenue, room 101 (the TEAL classroom). *

**4. The Math 225 town hall** is scheduled for Thursday, August 10, at 1pm ET. It will be run by Pablo Boixeda Alvarez (the instructor of Math 225), and Miki Havlickova (co-director of undergraduate studies in math). We will have a short presentation at the beginning, and then stay for questions, ending around 2pm. The session will be via zoom: click here to access it. (In case zoom asks for a passcode, it’s 808217 )

**5. Preference selection** for sections of calculus (Math 110, 112, 115, 120) will be open from 8am ET on Monday, August 8 until 5pm ET on Monday, August 14 . Please note that the timing of your entry is not taken into account in any way. In particular, you will not be disadvantaged if you submit your entry after the advising sessions. (You may not enter preference selection for two different levels of calculus, so it is important to settle on your placement in advance.)

**6. The a****cademic fair** will take place on August 28, 12:30 - 2pm ET. The Fair is a great place to ask questions about the mathematics major, courses at the 200 level or higher, mathematics community, etc. (Please note that advisers at the Fair cannot help with calculus placement - for placement advice at the 100 level, please attend one of the sessions listed above.)

Detailed general information about the Fair can be found on the Yale Academic fair website.

#### Notes about the calculus placement exam

- For detailed information about the test, visit our Calculus and placement exam FAQ. The test is typically open in July.
- All incoming students who wish to take calculus during their first two years at Yale should take the test during the summer, so that they can have the results available, and take advantage of August advising. This includes students who have taken AP Calculus - your scores will be taken into account, but you will still have to complete the exam.
- The test is not required for registration in math courses level 200+. If you wish to start in Math 225, for example, you can just register for it. For information about starting in level 200 courses, please be sure to read the section on “choosing your first math course(s)” below.
- The exam is available through Canvas, on a site called “Math placement exam [current year]”. The link to Canvas is https://canvas.yale.edu/ . All incoming first-year students are automatically enrolled on the site at the end of June. Rising sophomores, juniors, and seniors should contact math.dus@yale.edu to be added to the site.
- Results of the test are available within five minutes from submission. You can access them by clicking on the “placement results” tab. (If the tab does not load, try a different browser - Canvas does not interact well with certain browsers. If that does not help, please send a message to canvas@yale.edu for technical support, and cc math.dus@yale.edu .)
- If you would like to take a course that differs from your placement exam result, please attend our calculus advising session in August. The advisers will be happy to help you. Note that you must take the test first - advisers cannot give you placement without it. Detailed information about the session will be posted on this site by the middle of July.
- After receiving placement to take your first calculus course, pre-register for the class through Yale’s preference selection website. Preference selection is typically open in August (for the Fall term), and in December (for the Spring term).

#### Where to go with placement questions

If it is June, July, or early August: Please plan to attend our zoom placement advising session, listed in the event schedule above. This is the main event of the year, and we strongly encourage you to attend, as it is the best place to get placement advice. The event takes place before the end of August preference selection, so it will be in time for you to register for Fall (or Spring) calculus sections.

Outside of the main session, our individual advising resouces are limited. We will likely have a smaller advising session at the end of August; if so, it will be announced in the event schedule above.

During the rest of the year, please reach out to the course director of the course that you would like to take, to discuss your placement. (If you’re not sure which course, then you can just pick any of the course directors.) The list of course directors is on the bottom of this page.

Please note that the course directors have a lot to manage, particularly at the beginning of the year, and it’s unlikely that they will instantly be able to schedule an appointment with everyone who reaches out to them for individual advice about placement - we have over a thousand students in calculus every year! We encourage you to take the placement exam early in the summer, and attend the main August advising session, if at all possible, so that you’re sure to get the advice you need in a timely manner.

#### Checklist for enrolling in calculus

- Take the placement exam on Canvas. (You must do this even if you have AP scores.)
- If you have questions about the calculus courses or placement exam, check the calculus and placement exam FAQ.
- If you have questions about your placement, attend our calculus advising session in August. Schedule of the sessions will be posted on top of this site by the middle of July.
- If you need help with your placement and missed the advising session, write to the faculty member listed below as a contact for the course into which you were placed.
- Pre-register for your first calculus course via Preference selection, typically during the second week of August. Note that you must have placement in the course in order to enroll, and you may not enter preference selection for more than one calculus course at the same time.
- If you miss preference selection, or need to switch section after it runs, you will be able to do so once registration opens at the beginning of term. You can sign up for any section of the course where you have placement, if the section has open spots - if it does not, you can enter the automated waitlist. More information about waitlists can be found at on the Registrar’s website.

#### Checklist for enrolling in courses level 200+

- Read the “choosing your first math course(s)” section below for advice about where to get started. If you have questions about anything, you can stop by the Academic Fair at the end of August, visit our August DUS office hours (listed on the Math DUS site), or write to math.dus@yale.edu .
- If you choose to start in a course level 200+ (such as Math 225), you do NOT need to take the calculus placement exam. If you have the pre-requisites for your chosen course, you can simply sign up once registration opens.
- If you have already completed multivariable calculus, linear algebra, or other advanced math courses, the notes in “choosing your first math course(s)” will help guide you through the options. (This is especially important if your potential major(s) require math, as you will need to make arrangements about the requirements with the major’s DUS right away.)

### Math introductory courses

Click the items below to learn about the different introductory and intermediate math courses at Yale.

#### Differential calculus (Math 110-111 and 112)

**Math 112 **focuses on differential calculus, where the goal is to measure a function’s instantaneous rate of change (the derivative). First, we define limits, which allow us to talk rigorously about instantaneous changes. Then, we define the derivative and find many rules that simplify its computation. Next, we apply the derivative to better understand function behavior (allowing us to minimize cost or maximize profit in economic models). Finally, we develop strategies for reversing the derivative process to find antiderivative functions. These antiderivatives have surprising applications, like computing areas and modeling population growth.

**Math 112** requires the placement exam. Students typically take the course after completing pre-calculus.

Another option is the **Math 110/111** sequence, a two-semester course that integrates pre-calculus and differential calculus topics. Successful completion of Math 110 and 111 is equivalent to Math 112 in that it satisfies the same major and professional-school (e.g. pre-med) requirements; however a student completing Math 110 and 111 receives two course credits and two QR credits.

**Math 110/111** requires the placement exam. Some prior knowledge of pre-calculus is helpful.

#### Integral calculus and series (Math 115 and 116)

**Math 115 **investigates the mathematics of accumulation. With integral calculus, we use information about instantaneous rates (like the velocity of a moving object) to compute net changes (like change in position). Series convergence tests and Taylor series help us understand how calculators work, and various other topics (solids of revolution, parametric equations, polar coordinates) allow us to apply the tools of single variable calculus in broader contexts. Throughout the semester, we consider both exact and approximate solutions to problems and investigate the role of error.

**Math 115 and 116** both cover integral calculus, and both can be used as pre-requisites for any course that requires Math 115. Math 116 places emphasis on applications to biology, and is particularly suitable for biology and pre-medical students.

**Math 115 and 116** require the placement exam, or completion of Math 111 or 112 at Yale. Typically, students place into Math 115 after completing a differential calculus course such as AP Calculus AB.

#### Multivariable calculus (Math 120 and 121)

**Math 120** extends skills and knowledge you gain in single variable calculus to two and three variables. We will study how to graph surfaces and solids in three dimensions, differentiate and integrate functions of several variables, optimize functions of several variables (for example when maximizing profit in economics), we will learn how to integrate along curves and surfaces (for example when calculating how much energy a solar panel will generate while the sun is shining on it), and many other topics.

**Math 120 and 121** both cover multivariable calculus, and both can be used as pre-requisites for any course that requires Math 120. Math 121 places emphasis on applications to biology, and is particularly suitable for biology and pre-medical students. [Note for 2023-24: Math 121 will not be offered this year.]

**Math 120 and 121** require the placement exam or completion of Math 115 or 116 at Yale. Typically, students place into Math 120 after completing an integral calculus course, such as AP Calculus BC.

#### Linear algebra and introduction to proofs (Math 225, 226 and 222)

**Math 225** is a linear algebra course, and a part of our introductory sequence into the math major. In linear algebra, you will learn key language and concepts used throughout pure mathematics as well as in a wide variety of applications. Linear algebra starts by studying systems of equations with many variables, and it builds a detailed understanding of how to work in abstract n-dimensional space. This version of linear algebra focuses on concepts, and provides an introduction to writing mathematical proofs.

Math 225 usually has a zoom town hall meeting in early August: it will be listed under the “Summer / Fall event schedule” toward the top of this page.

We also offer an intensive version of this course, **Math 226**, for students who are looking for an extra challenge. Math 226 will teach the same main topics as Math 225, but it may go into more depth, ask more challenging problems on homework and exams, or cover optional side topics. Both courses prepare students equally well for subsequent mathematics courses, or for using proofs, problem solving skills, and linear algebra in other fields.

If you start in Math 226, and find that it is too time consuming, you can move to Math 225 until Midterm. You can read more about how moving between courses work in our Calculus FAQ.

**Math 222** is a linear algebra course best suited for students who wish to focus on applications and practical solving problem practice, rather than abstract mathematics and mathematical proofs. It is often taken by students majoring in engineering, technology, science, social sciences, and economics. Mathematics majors, as well as students majoring in Math + CPSC or Math + Econ, need to complete Math 225 or 226, rather than 222.

**Math 222, 225, and 226 **do not require the placement exam or preference selection, you can simply sign up for them when registration opens. They can be taken directly after integral calculus, though most students complete multivariable calculus first. For advice about whether linear algebra may be the best place for you to start, please see the section below on choosing your first math course(s) at Yale (particularly if you have not yet taken multivariable calculus, and it may be required by your prospective major or other courses you plan to take).

#### Real analysis (Math 255 and 256)

**Math 255** is an introduction to real analysis, and a part of our introductory sequence into the math major. The topics are similar to differential and integral calculus, but it’s done in a rigorous way that focuses on proofs and details of the concepts that happen behind the scenes in calculus. Said differently, our calculus sequence teaches you how to use the tools of limits, derivatives, and integrals to solve a variety of problems, whereas this course teaches you how to build all of those tools from scratch.

We also offer an intensive version of this course, **Math 256**, for students who are looking for an extra challenge. Math 256 will teach the same main topics as Math 255, but it may go into more depth, ask more challenging problems on homework and exams, or cover some optional side topics. Both courses prepare student equally well for subsequent math courses, or for using real analysis in other fields.

Math 255 and 256 require prior completion of Math 225 or 226.

Math 255 and 256 do not use the placement exam or preference selection. It requires Math 225 or 226. If you have taken a proof-based linear algebra course elsewhere, and are considering Math 255 or 256 as your first math course, please contact the Math DUS at math.dus@yale.edu to assist you with placement, as well as with requirements of your potential major (if it requires linear algebra). For more information, see section on choosing your first math course(s) below.

#### Multivariable analysis (Math 302)

**Math 302** is a rigorous and exciting course that explores some of the beautiful connections between linear algebra and real analysis. For this reason, it is intended to be taken after completing 225/226 and 255/256.

This course is an introduction to manifolds, which are low-dimensional shapes living in a higher dimensional universe (e.g., a one-dimensional coat hanger or a two-dimensional eggshell each living in a three-dimensional world), and it teaches how to do calculus in this abstract setting. Although the focus of this course will be on abstract mathematical objects and their properties, the material in this course also has applications to a broad range of topics including theoretical physics and machine learning.

Many students who have not completed multivariable calculus in high school would benefit from taking Math 120. However, prospective math majors, and students excited about rigorous proof-based math should consider taking Math 225/226, Math 255/256 and Math 302 instead. Math 302 is also appropriate for students who have already completed Math 120 and are curious to learn multivariable analysis rigorously.

Math 302 does not use the placement exam or preference selection. It requires both proof-based linear algebra, and real analysis (Math 225 and 255). If you have taken similar courses elsewhere and wish to start at Yale with Math 302, please be sure to contact the Math DUS during the summer before your first year, at math.dus@yale.edu , so we can help you as far as course placement as well as requirements of your potential major. For more information, please see the section on choosing your first math course(s) below.

#### Other 100-level courses (Math 118, 106, 107, 108)

**Math 118 **teaches a combination of multivariable calculus and linear algebra. It is suited for students majoring in Economics or Social sciences who wish to learn the most relevant mathematics in one semester. Students intending to take further courses in mathematics, or wishing to go to graduate school in their field, should consider multivariable calculus (Math 120) and linear algebra (Math 222, 225, or 226) instead.

Pre-requisite for Math 118 is differential calculus (such as Math 112 or AP Calculus AB). Knowledge of integral calculus is recommended.

The math department offers other Quantitative Reasoning (QR) courses that assume no calculus experience:

In **Math 107** (Mathematics in the real world), students use mathematical ideas to solve real world problems. Topics include compound interest, population growth, probability and its applications to games of chance, mortgage payments, false positives in drug testing, computer security, and other questions.

**Math 108** (Estimation and error) leads students through a problem-based investigation of basic mathematical principles and techniques that help make sense of the world. Applications include geology, ecology, finance, and other fields.

In Spring 2023, we anticipate having a new level 100 seminar similar to the three just above. It will be taught by Dr. Anderson, on the subject of **voting systems**.

#### Other 200-level courses (Math 241, 242, 244, 246, 270)

**Math 241 and 242:** probability and statistics teach us how to collect, analyze, and interpret data. That being so, it is directly useful in almost any field where experiments, surveys, or other means for data collection are employed.

**Math 244:** discrete mathematics includes many topics of interest in computer science.

**Math 246: **differential equations can be used for modeling complex systems in biology, physics, economics, and other fields.

**Math 270: **set theory is at the foundation of mathematics and often taken by philosophy students.

### Introductory course sequences

Click the items below to see how your first math course fits into our introductory sequences.

#### Calculus sequence

Many incoming students enroll in one of our calculus courses (Math 110, 112, 115, 116, 120, or 121). Within calculus, the sequence of courses is

- Math 110 + 111 or Math 112 (differential calculus, similar to AP Calculus AB)
- Math 115 or Math 116 (integral calculus, similar to AP Calculus BC)
- Math 120 or Math 121 (multivariable calculus). [Note that Math 121 is not offered in 2023-24.]

If you are considering taking calculus, you will need to complete our placement exam, so that we can help you determine the best place to start. We highly recommend taking the test during the summer before you first enroll at Yale. The result will be valid for two years.

#### Required introductory courses for the math major

Pre-requisite for the mathematics major is integral calculus (for example Math 115, or AP Calculus BC).

Following the pre-requisite, the introductory sequence includes:

- Linear algebra (Math 225 or 226),
- Real analysis (Math 255 or 256),
- Multivariable analysis or calculus (Math 302 or 120).

Most students complete multivariable calculus before enrolling in linear algebra.

Prospective mathematics majors and students with interest in abstract mathematics may consider enrolling in Math 225 (or 226) directly after integral calculus (e.g. Math 115 or AP Caclulus BC), and completing their vector analysis/multivariable calculus requirement with Math 302.

The placement exam is NOT required for any math course level 200 or above. In particular, you do not need it in order to enroll in Math 225, or 226. For information about starting in level 200 courses, please be sure to read the section on “choosing your first math course(s)” below.

### Choosing your first math course(s) at Yale

Click items below to see our recommendations; they are sorted by the major(s) you are considering, and each describes some options, depending on the math courses you have completed so far (whether at the level of pre-calculus, calculus, or more advanced courses such as linear algebra).

#### Mathematics, or Intensive mathematics

**If you have not completed a full integral calculus course**, including sequences and series, then you will need to complete the calculus placement exam, and start in the calculus sequence. For more information, see the above notes about the placement exam, and checklist for enrolling in calculus.

**If you have completed integral calculus but have not done multivariable calculus,** then the best course to start with is either Math 120, or Math 225 or 226. Note that Math 120 requires the placement exam; Math 225 and 226.

Most students with integral calculus background will start with Math 120, to finish the calculus sequence, and gain more experience with mathematics before enrolling in proof-based courses. This choice is recommended particularly if you may need multivariable calculus for another major you are considering, or if it is needed for other courses you might like to take soon (such as in physics or economics). After Math 120, you would move to Math 225 or 226, and Math 255 or 256, which would complete your introductory sequence requirement.

If you are excited to start with proof-based mathematics right away, then Math 225 or 226 may be the best course. For advice choosing between the two, see the above section on “Linear algebra and introduction to proofs”. Your next course would be Math 255 or 256, and you would complete the introductory sequence with Math 302 (or with Math 120, if you decide later that you prefer that).

**If you have completed multivariable calculus but have not taken linear algebra,** then the best choice is Math 225 or 226. For advice choosing between the two, see the above section on “Linear algebra and introduction to proofs”. Your next course would be Math 255 or 256, and you would complete the introductory sequence with Math 302. None of these courses require a placement exam.

(You will not need to have your multivariable calculus course evaluated by the math department, since it would not change the requirements in any way: all students have the option of satisfying the multivariable requirement with Math 302, whether or not they have already completed a course equivalent to Math 120.)

**If you have completed multivariable calculus and have taken linear algebra, **then the options are as follows:

First, we need to determine whether your linear algebra course was a theoretical, fully proof-based course intended for mathematics majors, or not. Most linear algebra courses are not, and chances are that you will still need to enroll in Math 225 or 226.

A proof-based course teaches techniques for writing mathematical proofs, and problem sets and exams focus mostly on proving abstract mathematics results. If the course explicitly taught students how to prove results by induction and by contradiction, and nearly every assigned problem had the phrase “prove [something]” in it, then the course was probably proof based; if not, then it probably wasn’t.

If you are not sure whether your linear algebra was proof-based or not, send a syllabus and either an exam or a problem set (the questions, not the solutions) to math.dus@yale.edu; we will be happy to take a look and advise you.

If the linear algebra course **was not** proof based, then you will still need to take Math 225 or 226. For advice choosing between the two, see the above section on “Linear algebra and introduction to proofs”. Your next course would be Math 255 or 256, and you would complete the introductory sequence with Math 302.

If the linear algebra course **was** proof based, then please reach out to the DUS during the summer before your first year, at math.dus@yale.edu . Yale does not permit students to get credit for courses taken during high school, but we can evaluate the course you have taken, and if it seems reasonably equivalent to Math 225, we will give you permission to substitute a higher level course in the same area for Math 225, in requirements of the math major. We will also help you find the best starting course (likely Math 255 or 256 – or higher, if you have taken real analysis as well). Please note that this permission has to be granted by the Math DUS - obtaining permission from the Math 255 instructor to enroll will not take care of requirements of the math major.

To evaluate the course, we will need a copy of the syllabus or another document with a detailed list of topics taught, a copy of your transcript (unofficial is fine), and a final exam for the course. If the instructor wishes to keep the test confidential, it can be sent directly to math.dus@yale.edu .

If you know proof-based linear algebra but have not taken officially taken a course in it, or if your course is not evaluated as equivalent to Math 225, you can try placing out of Math 225 by taking an exam during the summer before your first semester at Yale. Please reach out to math.dus@yale.edu before mid-August, and we will help you with the arrangements.

The exam takes place at the end of August, after everyone is on campus, usually the Tuesday before classes start. It is around 90 minutes long, and it is focused on proof writing, and the theoretical side of linear algebra (same way as Math 225).

Note that the test must be taken **before **you first enroll at Yale, and that it does not grant course credit – it may only allow you to substitute another course in the same area for Math 225.

To sign up for the exam, you must e-mail math.dus@yale.edu by August 15.

#### Math + CPSC

**If you have not completed a multivariable calculus course, **then you will need to complete the calculus placement exam, and start in the calculus sequence, through Math 120. For more information, see the above notes about the placement exam, and checklist for enrolling in calculus.

**If you have completed a multivariable calculus course** equivalent to Math 120, then please reach out to the math DUS during the summer before your first year, at math.dus@yale.edu . Yale does not permit students to get credit for courses taken during high school, but we can evaluate the course you have taken, and if it seems reasonably equivalent to Math 120, we will give you permission to substitute a higher level course in the same area for Math 120. We will also help you determine the best math course to start with. In most cases, this will be Math 225 or 226; but it could be a higher level course, if you have completed linear algebra already.

To evaluate the course, we will need a copy of the syllabus or another document with a detailed list of topics taught, a copy of your transcript (unofficial is fine), and a final exam for the course. If the instructor wishes to keep the test confidential, it can be sent directly to math.dus@yale.edu . When you send the materials, please be sure to mention that the evaluation is for the CPSC + Math major.

**To complete the specifically required courses** (other than the senior requirement), you will need to complete Math 225 or 226 (proof-based linear algebra), and Math 244 (discrete math). We recommend taking Math 225 or 226 before Math 244, as the former provides a thorough preparation in proof writing, and this is very helpful skill in Math 244.

If you have already taken some of these courses prior to enrolling at Yale, please reach out to the DUS at math.dus@yale.edu as soon as possible. We will work with you to evaluate the courses, and find the best possible options for you as far as requirements of the major.

**Other courses frequently taken by Math + CPSC** majors include Math 241 (probability theory), Math 246 (ordinary differential equations), Math 255 (analysis 1), Math 270 (set theory), Math 305 (analysis 2), Math 350 (abstract algebra). You can find more information about most of these courses in the above section on “Math introductory courses”.

#### Math + Economics

**If you have not completed a multivariable calculus course, **then you will need to complete the calculus placement exam, and start in the calculus sequence, through Math 120. For more information, see the above notes about the placement exam, and checklist for enrolling in calculus.

For Math + Econ majors, it is important to take multivariable calculus as soon as possible. We recommend starting with your Yale calculus classes right away, in the Fall of your first year.

**If you have completed a multivariable calculus course** equivalent to Math 120, then please reach out to the math DUS during the summer before your first year, at math.dus@yale.edu . Yale does not permit students to get credit for courses taken during high school, but we can evaluate the course you have taken, and if it seems reasonably equivalent to Math 120, we will give you permission to substitute a higher level course in the same area for Math 120 in requirements of the major. We will also help you determine the best math course to start with. In most cases, this will be Math 225 or 226; but it could be a higher level course, if you have completed linear algebra already.

To evaluate the course, we will need a copy of the syllabus or another document with a detailed list of topics taught, a copy of your transcript (unofficial is fine), and a final exam for the course. If the instructor wishes to keep the test confidential, it can be sent directly to math.dus@yale.edu . When you send the materials, please be sure to mention that the evaluation is for the Econ + Math major.

**To complete the specifically required courses** (other than the senior requirement), you will need to complete Math 225 or 226, and Math 255. Math 225 or 226 must be taken first.

If you have already taken some of these courses prior to enrolling at Yale, please reach out to the DUS at math.dus@yale.edu as soon as possible. We will work with you to evaluate the courses, and find the best possible options for you as far as requirements of the major.

**Other courses frequently taken by Math + Econ majors **include Math 244 (discrete math), Math 246 (differential equations), Math 270 (set theory), Math 330 (advanced probability). You can find more information about most of these courses in the above section on “Math introductory courses”.

#### Math + Phil

**If you have not completed a multivariable calculus course, **then you will need to complete the calculus placement exam, and start in the calculus sequence, through Math 120. For more information, see the above notes about the placement exam, and checklist for enrolling in calculus.

**If you have completed a multivariable calculus course** equivalent to Math 120, then please reach out to the DUS during the summer before your first year, at math.dus@yale.edu . Yale does not permit students to get credit for courses taken during high school, but we can evaluate the course you have taken, and if it seems reasonably equivalent to Math 120, we will give you permission to substitute a higher level course in the same area for Math 120. We will also help you determine the best math course to start with.

To evaluate the course, we will need a copy of the syllabus or another document with a detailed list of topics taught, a copy of your transcript (unofficial is fine), and a final exam for the course. If the instructor wishes to keep the test confidential, it can be sent directly to math.dus@yale.edu . When you send the materials, please be sure to mention that the evaluation is for the Phil + Math major.

While linear algebra is not required for the major, **we very much recommend completing Math 225 or 226,** partly for the topic they teach, and partly because they provide a thorough preparation with proof writing, which is an essential skill for the major.

**To complete the specifically required courses** (other than the senior requirement), you will need to complete Math 270. We recommend first completing Math 225 or 226, for the extra proof preparation, but this is not required.

**Other courses frequently taken by Math + Phil** majors include Math 241 (probability theory), Math 244 (discrete math), Math 246 (differntial equations), Math 255 (analysis 1). You can find more information about most of these courses in the above section on “Math introductory courses”.

#### Math + Phys

**If you have not completed a multivariable calculus course, **then you will need to complete the calculus placement exam, and start in the calculus sequence, through Math 120. For more information, see the above notes about the placement exam, and checklist for enrolling in calculus.

For Math + Phys majors, it is important to take multivariable calculus as soon as possible. We recommend starting with your Yale calculus classes right away, in the Fall of your first year.

**If you have completed a multivariable calculus course** equivalent to Math 120, then please reach out to the DUS during the summer before your first year, at math.dus@yale.edu . Yale does not permit students to get credit for courses taken during high school, but we can evaluate the course you have taken, and if it seems reasonably equivalent to Math 120, we will give you permission to substitute a higher level course in the same area for Math 120, in requirements of the major. We will also help you determine the best math course to start with.

To evaluate the course, we will need a copy of the syllabus or another document with a detailed list of topics taught, a copy of your transcript (unofficial is fine), and a final exam for the course. If the instructor wishes to keep the test confidential, it can be sent directly to math.dus@yale.edu . When you send the materials, please be sure to mention that the evaluation is for the Phys + Math major.

While linear algebra is not required for the major, **we very much recommend completing Math 222, 225 or 226,** as it teaches skills directly relevant to the major, as well as to many of the potential subsequent courses. For students who wish to keep open the possibility of taking proof-based courses in the future, Math 225 or 226 is highly recommended over Math 222, as the former provides a thorough preparation with proof writing.

**Other courses frequently taken by Math + Phys majors** include Math 241 (probability theory), Math 244 (discrete math), Math 246 (differntial equations), Math 255 (analysis 1). Students interested in theoretical physics and proof-based courses often include some of the level 300 courses, most often Math 305 (analysis 2), Math 310 (complex analysis), or Math 350 (abstract algebra). You can find more information about most of these courses in the above section on “Math introductory courses”.

#### Other majors

For majors other than the mathematics ones listed above, we share below some notes that may help you get started. We would also encourage you to speak with your (intended) major’s DUS - they are the best person to advise you about courses to take.

If you have completed some of your major’s requied math courses prior to enrolling at Yale, please reach out to the major’s DUS for advice, and to ask what options may be available as far as completing the requirements. (The math DUSes are happy to provide advice, but any arrangements about requirements of another major must be settled with the major’s DUS, rather than with the Math department.)

Now the notes:

Most Yale students complete a calculus course at some point during their studies. If you have not yet completed the calculus sequence all the way through multivariable, and you either wish to continue with the sequence, or you want to keep your options open as far majors or other courses that require calculus, then one of the courses in the calculus sequence may be a great place to start.

In order to enroll, you will need to complete the placement exam, and then register for a section of your course through preference selection. For details, see the above notes about the calculus sequence, placement exam, and checklist for enrolling in calculus.

If you have never taken calculus, do not expect to need calculus for your major or other classes, and would prefer to take another kind of math course at Yale, then you might enjoy one of our level 100 seminars (numbered 101 - 109): these are described above under “Other courses at the 100 level”. The seminars have limited enrollment, and do not expect any prior knowledge of calculus. They teach a combination of math topics and problem solving skills that can be useful for students in any major.

If you have completed a multivariable calculus course, and your major **does not **require calculus, then you can enroll directly in a higher level math course. You do not need to have your previous math courses evaluated: for purposes of pre-requisite to other courses, you can simply tell the instructor that you have completed the relevant course prior to enrolling at Yale.

For courses to start with, after multivariable: Frequent choices for many majors include linear algebra (Math 222, 225, or 226), probability theory (Math 241), discrete mathematics (Math 244), or differential equations (Math 246). You will find these described under “Linear algebra” and “Other courses at the 200 level” above. If you need any help choosing, please don’t hesitate to stop by the academic fair, write to math.dus@yale.edu, stop by one of our August office hours (to be posted on the Math DUS website,) or consult the DUS of your major for recommendations.

If you have completed a multivariable calculus course and your major **does** require calculus, then you will need to meet with the DUS of your major to discuss what the options may be. Some majors allow students to substitute a higher level math course, for example. You will not need to have the course evaluated by the Math department: options regarding the requirements must be done with the DUS of the major. We are happy to help you decide which math course to start with, however, so please feel free to stop by or write to math.dus@yale.edu for advice.

If you would like more assistance with these choices, the best place to get advice is the Academic Fair at the end of August, or by e-mail at math.dus@yale.edu, or via zoom by stopping by one of our August office hours (to be posted on the Math DUS site).

### Contact information

#### Where to go with additional questions

With additional questions, a good place to start are our FAQ’s:

If your question is not addressed there, please contact the Math DUS at math.dus@yale.edu . Please allow at least two weekdays for an answer. (During school recess, especially winter and summer break, our response time may be longer.)

For questions about the calculus courses, please contact the course director listed below. For single-section courses, the best person to contact is the instructor.

#### Course directors for 2024-25

- Math 110 and 111: John Hall
- Math 112: Meghan Anderson (Fall), C.J. Argue (Spring) [e-mail starts with cj.argue]
- Math 115: Brett Smith (Fall and Spring)
- Math 120: Andrew Yarmola (Fall), Maria Siskaki (Spring)
- For questions about a specific single-section course, the best person to ask is the instructor (listed in Yale Course Search).
- For questions about calculus placement: If it is prior to April or December registration, please contact one of the course directors listed above. During the summer, please attend the calculus advising session in early August (listed in the schedule toward the top of this page).
- For other advice (e.g. about level 200+ course selection or requirements of the major), please don’t hesitate to reach out to math.dus@yale.edu

#### .Peer liaisons with Student Accessibility Services

The Student Accessibility Services (SAS) Office works with students with ADHD/Autism/learning difficulties, sensory or motor disabilities, mental health concerns, temporary disabilities like concussions and serious injuries, and more. Our peer liaison program is designed so that first-years can connect with someone who has first-hand experience with navigating accommodations at Yale. If you think you could benefit from additional resources in regards to academics and general college life, please don’t hesitate to fill out this form and the PL for your college will reach out ASAP.